On continuous extension of locally homeomorphic simplicial maps of \(\mathbb{R}^2\) into itself by \(\sigma\)-processes (Q1360817)

In [\textit{F. Dell'Accio}, Math. Notes 58, No. 3, 989-992 (1995); translation from Mat. Zametki 58, No. 3, 452-455 (1995; Zbl 0868.58012)], the author considered a conjecture of Vitushkin concerning the Jacobi problem and constructed a covering \(M\) of \(\mathbb{R}^2\). This article is a supplement containing additional properties of this covering. The surface \(M\) can be glued from a regular hexagon without vertices according to the word \(abca^{-1}bc\), thus \(M\) is a Klein bottle with two points removed. The author shows that \(M\) can be embedded in the compactification of \(\mathbb{R}^2\) by a finite tree, and studies the extension of the covering \(M\to \mathbb{R}^2\) to this compactification. The proof is obtained by combinatorial study of certain trees of circles.